Nuprl Lemma : pcw-pp-barred-W
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[pp:n:ℕ × (ℕn ⟶ cw-step(A;a.B[a]))]. (Barred(pp) ∈ ℙ)
Proof
Definitions occuring in Statement :
cw-step: cw-step(A;a.B[a]),
pcw-pp-barred: Barred(pp),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
pcw-pp-barred: Barred(pp),
prop: ℙ,
and: P ∧ Q,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
cw-step: cw-step(A;a.B[a]),
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
spreadn: spread3,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
ext-family: F ≡ G,
ext-eq: A ≡ B,
pi1: fst(t)
Lemmas referenced :
less_than_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel-alt,
lelt_wf,
cw-step_wf,
param-co-W-ext,
unit_wf2,
it_wf,
assert_wf,
isr_wf,
equal_wf,
nat_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
productElimination,
thin,
productEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
applyEquality,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
minusEquality,
cumulativity,
lambdaEquality,
functionExtensionality,
hypothesis_subsumption,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
functionEquality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[pp:n:\mBbbN{} \mtimes{} (\mBbbN{}n {}\mrightarrow{} cw-step(A;a.B[a]))]. (Barred(pp) \mmember{} \mBbbP{})
Date html generated:
2017_04_14-AM-07_43_32
Last ObjectModification:
2017_02_27-PM-03_14_12
Theory : co-recursion
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